Doctoral Thesis

DOI

10.11606/T.43.2000.tde-28112013-102436

Document

Author

Full name

Júlio César Bastos de Figueiredo

Institute/School/College

Knowledge Area

Date of Defense

Published

São Paulo, 2000

Supervisor

Committee

Malta, Coraci Pereira (President)

Caldas, Ibere Luiz

Furuie, Sergio Shiguemi

Koiller, Jair

Ranvaud, Ronald Dennis Paul Kenneth Clive

Caldas, Ibere Luiz

Furuie, Sergio Shiguemi

Koiller, Jair

Ranvaud, Ronald Dennis Paul Kenneth Clive

Title in Portuguese

Equações Diferenciais não Lineares com Três Retardos: Estudo Detalhado das Soluções

Keywords in Portuguese

Equações diferenciais não lineares

Física teórica

Física teórica

Abstract in Portuguese

In this thesis we study the behavior of a simple control system based on a delay differential equation with multiple loops of negative feedback. Numerical solutions of the delay differential equation with N delays d/dt x(t) = -x(t) + 1/N POT.N IND.i=1 / POT.n IND.i + x (t- IND.i) POT.n have been investigated as function of its parameters: n, i and i. A simple numerical method for determine the stability regions of the equilibrium points in the parameter space (i, n) is presented. The existence of a doubling period route to chaos in the equation, for N = 3, is characterized by the construction of bifurcation diagram with parameter n. A numerical method that uses the analysis of Poincaré sections of the reconstructed attractor to find aperiodic solutions in the parameter space of the equation is also presented. We apply this method for N = 2 and get evidences for the existence of chaotic solutions as result of a period doubling route to chaos (chaotic solutions for N = 2 in that equation had never been observed). Finally, we study the solutions of a piecewise constant equation that corresponds to the limit case n .

Title in English

Nonlinear differential equations with three delays: detailed study of the solutions.

Keywords in English

Nonlinear differential equations

Theoretical physics

Theoretical physics

Abstract in English

In this thesis we study the behavior of a simple control system based on a delay differential equation with multiple loops of negative feedback. Numerical solutions of the delay differential equation with N delays d/dt x(t) = -x(t) + 1/N POT.N IND.i=1 / POT.n IND.i + x (t- IND.i) POT.n have been investigated as function of its parameters: n, i and i. A simple numerical method for determine the stability regions of the equilibrium points in the parameter space (i, n) is presented. The existence of a doubling period route to chaos in the equation, for N = 3, is characterized by the construction of bifurcation diagram with parameter n. A numerical method that uses the analysis of Poincaré sections of the reconstructed attractor to find aperiodic solutions in the parameter space of the equation is also presented. We apply this method for N = 2 and get evidences for the existence of chaotic solutions as result of a period doubling route to chaos (chaotic solutions for N = 2 in that equation had never been observed). Finally, we study the solutions of a piecewise constant equation that corresponds to the limit case n .

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30820Figueiredo.pdf (1.54 Mbytes)

Publishing Date

2013-12-18

Centro de Informática de São Carlos

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